Known systems for precision alignment or displacement measurements have a number of common drawbacks. In particular, such systems are generally complex and expensive. Additionally, many such systems are inflexible in requirements, e.g., space and/or isolation requirements, making implementations awkward or impossible in many applications. Many require specific patterns such as grating patterns to be laid-down on the object being measured to produce moiré or diffraction patterns. Such patterns can be highly regular, so that spatial-uniqueness (or false matches) can become an issue. Also many precision measurement systems that are accurate at small dimensions are specifically designed for alignment sensing and cannot track movement or provide quantitative displacement information. Further, the systems that do provide quantitative displacement information are often unable to do so in real-time because of required scanning processes or significant post-processing.
Current measurement systems for tracking of an object can be broadly categorized as being optical or non-optical measurement systems. An interferometer is one example of an optical measurement system that can precisely measure the position or velocity of an object by interfering or comparing a beam reflected from the object with a reference beam. Other optical interference based measurement systems are known that track object movement by measuring the movement of diffraction patterns that gratings mounted on the object generate. Some other optical measurement systems use image correlations to detect the alignment or movement of known geometric patterns. Non-optical techniques are also available or proposed for tracking object movement. Examples of non-optical systems for precise measurements of small displacements include a Scanning Electron Microscope (SEM), an Atomic Force Microscope (AFM), or a capacitance sensing system.
An advantage of optical measurement systems when compared to the non-optical systems is the availability of precise and relatively inexpensive beam sources and optical elements. Accordingly, optical systems for alignment or tracking have been implemented at scales ranging from tracking astronomical bodies to tracking missiles to tracking integrated circuit structures.
One specific optical technique for measuring a displacement uses Fourier transforms of consecutive images of an object. A well-known property of Fourier transforms is that a position shift in an image results in a phase delay in the Fourier transform of the image. This property for a two-dimensional Fourier transform is expressed in Equation 1, where functions ƒ(x,y) and ƒ(x−x0, y−y0) can respectively represent intensity variations of an image and a shifted image and a function F(ωx,ωy) represents the Fourier transform of function ƒ(x,y). In the Fourier transform of Equation 1, the phase (ωxx0+ωyy0) is a linear function of frequencies (ωx,ωy) having slopes equal to the displacements (x0, y0) of the image. However, determination of the displacement vector through phase delays in measurement systems have generally required transformations back to the spatial domain because comparisons of phases of transformed functions in the frequency domain correspond to subtractions of measurements that are believed to increase the effects of measurement noise.                Equation 1:ƒ(x−x0,y−y0)ei(ωxx0+ωyy0)F(ωx,ωy)        
Optical systems for tracking movement with nanometer scale accuracies are desired, particularly for manufacturing of nanometer-scale devices.